Topology of Liouville foliations in the Steklov and the Sokolov integrable cases of Kirchhoff's equations

Abstract

Elastic relaxation in dry-etched periodic wires fabricated from molecular beam epitaxy grown Si/SiGe multilayers was studied by coplanar and grazing incidence (GID) high-resolution x-ray diffraction. The inhomogeneous strain distribution in the wires was calculated by the finite element method, which provided the input data for simulations of the scattered intensities using kinematical diffraction theory used for comparison with measured reciprocal space maps. A fabrication-induced layer covering the wire surfaces, modifies the strain distribution. Using GID, the geometrical shape of the wires and their in-plane strain can be determined independently of each other.

Topological classification of Liouville foliations for the Kovalevskaya integrable case on the Lie algebra so(3, 1)

“…We will use the information about some previously studied integrable cases, namely the classical Kovalevskaya case and the Sokolov integrable case studied in [16] and [14]. The first case is the limit of the Kovalevskaya case on so(3, 1) when κ → −0.…”

“…The cycle λ β 4 on tori of the family (6) (i.e. λ β 1 on family I of tori in [14]) and λ β 4 on tori of the family (7) (i.e. λ β 4 on family V of tori in [14]) are expressed similarly by a choose of basis of the lattice.…”

“…Therefore the admissible coordinate systems for the arcs α 3 , β 4 , γ 10 , γ 11 , δ 3 , ξ 6 are known from [14].…”

Singularities of integrable Liouville systems, reduction of integrals to lower degree and topological billiards: Recent results

“…The integrable systems by Kovalevskaya [1], Kovalevskaya-Yahya [11], Kovalevskaya on the Lie algebra (4) [12], Goryachev-Chaplygin-Sretensky [5], Sokolov [13], Dullin-Matveev [14] with the integrals of degrees 3 and 4 are modeled (that is, are piecewise smooth Liouville equivalent) in the suitable energy zones (that is, on the suitable constantenergy 3-manifolds) by the integrable topological billiards with the canonical integral of degree 2. In other words, the integrals of higher degrees are reduced to the one and the same quadratic integral…”

“…In the first column, there are corresponding billiards, in the second column there are corresponding Fomenko-Zieschang invariants for these integrable systems, in the third column the corresponding cases of integrability are indicated. Here the numbering of the invariants and the isoenergy zones which are indicated in brackets are taken from the papers [1,5,[11][12][13][14]. In the fourth column, the topological type of the corresponding isoenergy 3-manifold is presented.…”